{
  "abstract": "We study the similarity problem for representations of compact and discrete quantum groups. We first prove that every non-degenerate contractive representation of a compact or discrete quantum group is automatically unitary. Then, for compact quantum groups, we develop an interpolation method for constructing explicit non-unitarizable non-degenerate representations with norms arbitrarily close to $1$. In particular, this applies to all non-Kac compact quantum groups whose dual has subexponential growth, including the Drinfeld--Jimbo quantum groups $G_q$, as well as to the non-Kac free unitary quantum groups $U_F^+$ with $F\\in\\operatorname{GL}_2(\\mathbb{C})$. On the discrete quantum group side, we establish a quantum analogue of the classical lifting principle for non-unitarizable uniformly bounded representations. Combining this lifting principle with recent results on maximal Kac quantum subgroups, we construct explicit non-unitarizable non-degenerate representations with norms arbitrarily close to $1$ for all unitary free quantum groups $\\mathbb FU_F$ and all non-amenable orthogonal free quantum groups $\\mathbb FO_F$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27737",
  "author_total": 2,
  "authors": [
    {
      "name": "Fatemeh Khosravi"
    },
    {
      "name": "Sang-Gyun Youn"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0zh05gp483aak.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27737",
  "fulltext_url": null,
  "id": "0zh05gp483aak",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27737",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7204831573",
  "pdf_url": "https://arxiv.org/pdf/2608.27737",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0zh05gp483aak.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.27737",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27737"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Non-unitarizable representations of compact and discrete quantum groups",
  "title_text": "Non-unitarizable representations of compact and discrete quantum groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/0zh05gp483aak",
  "venue": "arXiv",
  "versions": []
}